In a certain factory, each day the expected number of accidents is related to the number of overtime hours by a linear equation. Suppose that on one day there were 1000 overtime hours logged and 8 accidents reported, and on another day there were 400 overtime hours logged and 5 accidents. What are the expected numbers of accidents when no overtime hours are logged?

Question

Let $ n(A) = m $ and $ n(B) = n $, if the number of subsets of $ A $ is 56 more than that of subsets of $ B $, then $ m + n $ is equal to:

Answer 1

Think of this as a proportion problem instead of writing a formula. Between the two recorded days, overtime hours dropped from 1000 to 400, a fall of 600 hours, and accidents dropped from 8 to 5, a fall of 3. So every drop of 600 overtime hours brings 3 fewer accidents, a fall of 1 accident for every 200 hours.

Now walk the same line back from the point (400 hours, 5 accidents) down to 0 hours, a further drop of 400 hours. At the rate of 1 accident less per 200 hours, dropping 400 hours means the accident count falls by $\frac{400}{200} = 2$. So the accidents at 0 hours are $5 - 2 = 3$.

2: too low, this would mean the count fell by 3, not 2, over the last 400 hours, which breaks the fixed rate found above.
3: matches the rate exactly, this is the correct value.
4: too high, it would mean accidents barely change with overtime, which does not match the drop from 8 to 5.
5: this is the accident count already recorded at 400 hours, so it cannot also be the value at 0 hours since the line is strictly decreasing.

Let's summarize:

The two data points fix one straight line, so the rate of change (1 accident per 200 hours) stays constant everywhere on it.
Walking back 400 more hours from the (400, 5) point at that rate gives the value at zero hours.

So the expected number of accidents with no overtime hours logged is 3.

Answer 2

Think of this as a proportion problem instead of writing a formula. Between the two recorded days, overtime hours dropped from 1000 to 400, a fall of 600 hours, and accidents dropped from 8 to 5, a fall of 3. So every drop of 600 overtime hours brings 3 fewer accidents, a fall of 1 accident for every 200 hours.

Now walk the same line back from the point (400 hours, 5 accidents) down to 0 hours, a further drop of 400 hours. At the rate of 1 accident less per 200 hours, dropping 400 hours means the accident count falls by $\frac{400}{200} = 2$. So the accidents at 0 hours are $5 - 2 = 3$.

2: too low, this would mean the count fell by 3, not 2, over the last 400 hours, which breaks the fixed rate found above.
3: matches the rate exactly, this is the correct value.
4: too high, it would mean accidents barely change with overtime, which does not match the drop from 8 to 5.
5: this is the accident count already recorded at 400 hours, so it cannot also be the value at 0 hours since the line is strictly decreasing.

Let's summarize:

The two data points fix one straight line, so the rate of change (1 accident per 200 hours) stays constant everywhere on it.
Walking back 400 more hours from the (400, 5) point at that rate gives the value at zero hours.

So the expected number of accidents with no overtime hours logged is 3.

Over Number	Run Rate
N-2	8.00
N	7.43
N+2	8.11
N+4	8.45
N+6	8.08

Show Hint

The Correct Option is B

Solution and Explanation

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